In a cooperative interaction, the environmental noise is usually expected to lead to a detrimental impact on the emergence and stability of cooperation [1]}, [2]} and thus requires additional supporting mechanisms such as apology and forgiveness [3]}, [4]}. Surprisingly, we have shown here that the presence of some noise that causes errors when deciding to participate in a prior commitment, can stabilise commitment compliance and cooperation, enabling ACD to become an ESS and risk-dominant against all other strategies. A non-negligible level of noise enables ACD to break ties with the commitment-accepting unconditional cooperators (ACC), who cooperate even when a commitment is not formed and thus can be easily exploited by non-committing defectors. In SI, we also considered execution noise that happens during the PD game. We showed that it has insignificant effects on the evolutionary dynamics and ESS analysis (e.g. none of the strategies can be an ESS for this type of noise).
